Functional SPDE with Multiplicative Noise and Dini Drift

TL;DR

This paper establishes existence, uniqueness, and non-explosion of solutions for a class of semi-linear functional SPDEs with multiplicative noise and Dini continuous drifts, and derives key inequalities in the finite-dimensional delay setting.

Contribution

It introduces new results on the well-posedness and inequalities for functional SPDEs with Dini drifts and multiplicative noise, expanding understanding in this complex area.

Findings

Proved existence and uniqueness of solutions

Derived log-Harnack inequality and $L^2$-gradient estimate

Analyzed the path space of the solution in delay setting

Abstract

Existence, uniqueness and non-explosion of the mild solution are proved for a class of semi-linear functional SPDEs with multiplicative noise and Dini continuous drifts. In the finite-dimensional and bounded time delay setting, the log-Harnack inequality and $L^2$-gradient estimate are derived. As the Markov semigroup is associated to the functional (segment) solution of the equation, one needs to make analysis on the path space of the solution in the time interval of delay.